What can people taking Coumadin (warfarin) eat without any side effects?

Most anything. Greens ( spinich, kale, mustard, turnip greens) can blunt the effect on the coumdain on thinning your blood. The important thing is, if you eat certain foods reguarly, you will not have wild swings on your pt. If you suddenly binge on foods such as these, that is a different story. And the taking of antibiotics can increase your pt.
Talk to a dietician. A good resource for information on what to eat when you are taking warfarin is a dietician. The web also has a wealth of information. One source is http://www.Livestrong.Com/article/488328-what-type-of-foods-can-i-eat-while-taking-warfarin/ good luck!